Let a - b - c be three non coplanar vectors- Define v 1- v 2- v 3 bywhich of the following statements is false-A- a - v 1- b - v 2- c - v 3-0B- v 1-v-2-v-3-v-2-v-3-v-1-v-3-v-1-v-2-3-a-b-c-C- -v-1-v-2v-2-v-3v-3-v-1-1-b-c-2-D- -v1v2v3-1-a b-c-

The correct option is A a⃗.v⃗_⃗1⃗+b⃗.v⃗_⃗2⃗+c⃗.v⃗_⃗3⃗=0 Using [V̅_̅1̅V̅_̅2̅V̅_̅3̅]=[a̅×b̅b̅×c̅c̅×a̅]/[a̅b̅c̅]= [a̅b̅c̅]^2/[a̅b̅c̅]^3=1/[a̅b̅c̅] a̅.v̅_̅1̅+b̅.v ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Union of Sets

Let a , b , c...

Question

Let $¯ a, ¯ b, ¯ c$ be three non coplanar vectors. Define $_{1},_{2},_{3}$ by
$_{1} = \frac{\to b \times \to c}{[\to a \to b \to c]},_{2} = \frac{\to c \times \to a}{[\to a \to b \to c]},_{3} = \frac{\to a \times \to b}{[\to a \to b \to c]}$
which of the following statements is false?

\to a ._{1} + \to b ._{2} + \to c ._{3} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

_{1} . (_{2} \times_{3}) +_{2} . (_{3} \times_{1}) +_{3} . (_{1} \times_{2}) = \frac{3}{[\to a \to b \to c]}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

[_{1}_{2}_{3}] = \frac{1}{[\to a \to b \to c]}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

[_{1} \times_{2}_{2} \times_{3}_{3} \times_{1}] = \frac{1}{[\to a \to b \to c]^{2}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\to a ._{1} + \to b ._{2} + \to c ._{3} = 0$

Using
$[_{1}_{2}_{3}] = \frac{[¯ a \times ¯ b ¯ b \times ¯ c ¯ c \times ¯ a]}{[¯ a ¯ b ¯ c]} = \frac{[¯ a ¯ b ¯ c]^{2}}{[¯ a ¯ b ¯ c]^{3}} = \frac{1}{[¯ a ¯ b ¯ c]} ¯ a ._{1} + ¯ b ._{2} + ¯ c,_{3} = \sum \frac{¯ a . ¯ b \times ¯ c}{[¯ a ¯ b ¯ c]} = 3, [v_{1} v_{2} v_{3}] = \frac{1}{[¯ a ¯ b ¯ c]} \sum [v_{1} v_{2} v_{3}] = \frac{[a b c]^{2}}{[a b c]^{3}}$

Suggest Corrections

Similar questions

Q. Consider three vectors

{\to v}_{1}, {\to v}_{2}

and

{\to v}_{3}

such that

{\to v}_{1} = {\to v}_{2} - {\to v}_{3}

where

{\to v}_{1} = (\to a \times ˆ i) \times ˆ i

{\to v}_{2} = (\to a \times ˆ j) \times ˆ j

{\to v}_{3} = (\to a \times ˆ k) \times ˆ k

.
(

\to a

is non zero vector), then

Q. If

{\to V}_{1} =^i - 2^j + 3^k

and

{\to V}_{2} = a^i + b^j + c^k

where

a, b, c \in {- 1, 0, 1, 2, 3}

then find the numbers of non-zero vectors

{\to V}_{2}

which are normal to

{\to V}_{1}

Q. The angle between

{\to V}_{1} =^i + 2^j + 2^k

and

{\to V}_{2} = 2^i + 3^j - 4^k

is:

Q. An impulse

\to 1

changes the velocity of a particle from

{\to v}_{1}

{\to v}_{2}

. Kinetic energy gained by the particle is

Two particles of masses $m_{1} a n d m_{2}$ in projectile motion have velocities ${\to v}_{1} a n d {\to v}_{2}$ respectively at time t = 0 and that is the moment when they collide. Their velocities become ${\to v}_{1}^{'} a n d {\to v}_{2}^{'}$ at time $2 t_{0}$ while still moving in air. The value of magnitude of change in total momentum would be

Join BYJU'S Learning Program

Let ¯a,¯b,¯c be three non coplanar vectors. Define →v1,→v2,→v3 by →v1=→b×→c[→a→b→c],→v2=→c×→a[→a→b→c],→v3=→a×→b[→a→b→c] which of the following statements is false?

The correct option is A →a.→v1+→b.→v2+→c.→v3=0 Using [¯V1¯V2¯V3]=[¯a×¯b¯b×¯c¯c×¯a][¯a¯b¯c]=[¯a¯b¯c]2[¯a¯b¯c]3=1[¯a¯b¯c]¯a.¯v1+¯b.¯v2+¯c,¯v3=∑¯a.¯b×¯c[¯a¯b¯c]=3,[v1v2v3]=1[¯a¯b¯c]∑[v1v2v3]=[abc]2[abc]3

Let $¯ a, ¯ b, ¯ c$ be three non coplanar vectors. Define $_{1},_{2},_{3}$ by
$_{1} = \frac{\to b \times \to c}{[\to a \to b \to c]},_{2} = \frac{\to c \times \to a}{[\to a \to b \to c]},_{3} = \frac{\to a \times \to b}{[\to a \to b \to c]}$
which of the following statements is false?